*Students are expected to know the following:*

*Students are expected to be able to do the following:*

## Big Ideas

## Big Ideas

Algebra allows us to generalize

*Sample questions to support inquiry with students:*- After solving a problem, can we extend it? Can we generalize it?
- How can we take a contextualized problem and turn it into a mathematical problem that can be solved?
- How do we tell if a mathematical solution is reasonable?
- Where can errors occur when solving a contextualized problem?
- What are the similarities and differences between quadratic functions and linear functions? How are they connected?
- What do we notice about the rate of change in a quadratic function?
- How do the strategies for solving linear equations extend to solving quadratic, radical, or rational equations?
- What is the connection between domain and extraneous roots?

The meanings of, and connections

*Sample questions**to support inquiry with students:*- How are the different operations (+, -, x, ÷, exponents, roots) connected?
- What are the similarities and differences between multiplication of numbers, powers, radicals, polynomials, and rational expressions?
- How can we verify that we have factored a trinomial correctly?
- How can visualization support algebraic thinking?
- How can patterns in numbers lead to algebraic generalizations?
- When would we choose to represent a number with a radical rather than a rational exponent?
- How do strategies for factoring x
^{2}+bx+cextend to ax^{2}+bx + c, a≠1 - How do operations on rational numbers extend to operations with rational expressions?

Quadratic relationships

*Sample questions to support inquiry with students:*- What are some examples of quadratic relationships in the world around us, and what are the similarities and differences between these?
- Why are quadratic relationships so prevalent in the world around us?
- How does the predictable pattern of linear functions extend to quadratic functions?
- Why is the shape of a quadratic function called a parabola?
- How can we decide which form of a quadratic function to use for a given problem?
- What effect does each term of a quadratic function have on its graph?

Trigonometry involves using proportional reasoning

- comparisons of relative size or scale instead of numerical difference

- using measurable values to calculate immeasurable values (e.g., calculating the width of a river using the distance between two points on one shore and an angle to a point on the other shore)
*Sample questions to support inquiry with students:*- How is the cosine law related to the Pythagorean theorem?
- How can we use right triangles to find a rule for solving non-right triangles?
- How do we decide when to use the sine law or cosine law?
- What would it mean for an angle to have a negative measure? Identify a context for making sense of a negative angle.

## Content

## Content

real number

- classification

powers

- positive and negative rational exponents
- exponent laws
- evaluation using order of operations
- numerical and variable bases

radical

- simplifying radicals
- ordering a set of irrational numbers
- performing operations with radicals
- solving simple (one radical only) equations algebraically and graphically
- identifying domain restrictions and extraneous roots of radical equations

polynomial factoring

- greatest common factor of a polynomial
- trinomials of the form ax
^{2}+ bx + c - difference of squares of the form a
^{2}x^{2}- b^{2}y^{2} - may extend to a(f(x))
^{2}+ b(f(x)) +c, a^{2}(f(x))^{2}- b^{2}(f(x))^{2}

rational

- simplifying and applying operations to rational expressions
- identifying non-permissible values
- solving equations and identifying any extraneous roots

quadratic

- identifying characteristics of graphs (including domain and range, intercepts, vertex, symmetry), multiple forms, function notation, extrema
- exploring transformations
- solving equations (e.g., factoring, quadratic formula, completing the square, graphing, square root method)
- connecting equation-solving strategies
- connecting equations with functions
- solving problems in context

linear and quadratic inequalities

- single variable (e.g., 3x - 7 ≤ -4, x
^{2}- 5x + 6 > 0) - domain and range restrictions from problems in situational contexts
- sign analysis: identifying intervals where a function is positive, negative, or zero
- symbolic notation for inequality statements, including interval notation

trigonometry

- use of sine and cosine laws to solve non-right triangles, including ambiguous cases
- contextual and non-contextual problems
- angles in standard position:
- degrees
- special angles, as connected with the 30-60-90 and 45-45-90 triangles

- unit circle
- reference and coterminal angles
- terminal arm
- trigonometric ratios
- simple trigonometric equations

financial literacy

- compound interest
- introduction to investments/loans with regular payments, using technology
- buy/lease

## Curricular Competency

## Curricular Competency

#### Reasoning and modelling

Develop thinking strategies

- using reason to determine winning strategies
- generalizing and extending

Explore, analyze

- examine the structure of and connections between mathematical ideas (e.g., trinomial factoring, roots of quadratic equations)

- inductive and deductive reasoning
- predictions, generalizations, conclusions drawn from experiences (e.g., with puzzles, games, and coding)

- graphing technology, dynamic geometry, calculators, virtual manipulatives, concept-based apps
- can be used for a wide variety of purposes, including:
- exploring and demonstrating mathematical relationships
- organizing and displaying data
- generating and testing inductive conjectures
- mathematical modelling

- manipulatives such as algebra tiles and other concrete materials

Estimate reasonably

- be able to defend the reasonableness of an estimated value or a solution to a problem or equation (e.g., the zeros of a graphed polynomial function)

- includes:
- using known facts and benchmarks, partitioning, applying whole number strategies to rational numbers and algebraic expressions
- choosing from different ways to think of a number or operation (e.g., Which will be the most strategic or efficient?)

Model

- use mathematical concepts and tools to solve problems and make decisions (e.g., in real-life and/or abstract scenarios)
- take a complex, essentially non-mathematical scenario and figure out what mathematical concepts and tools are needed to make sense of it

- including real-life scenarios and open-ended challenges that connect mathematics with everyday life

Think creatively

- by being open to trying different strategies
- refers to creative and innovative mathematical thinking rather than to representing math in a creative way, such as through art or music

- asking questions to further understanding or to open other avenues of investigation

#### Understanding and solving

Develop, demonstrate, and apply conceptual understanding of mathematical ideas through play, story, inquiry

- includes structured, guided, and open inquiry
- noticing and wondering
- determining what is needed to make sense of and solve problems

Visualize

- create and use mental images to support understanding
- Visualization can be supported using dynamic materials (e.g., graphical relationships and simulations), concrete materials, drawings, and diagrams.

Apply flexible and strategic approaches

- deciding which mathematical tools to use to solve a problem
- choosing an effective strategy to solve a problem (e.g., guess and check, model, solve a simpler problem, use a chart, use diagrams, role-play)

- interpret a situation to identify a problem
- apply mathematics to solve the problem
- analyze and evaluate the solution in terms of the initial context
- repeat this cycle until a solution makes sense

Solve problems with persistence and a positive disposition

- not giving up when facing a challenge
- problem solving with vigour and determination

Engage in problem-solving experiences connected

- through daily activities, local and traditional practices, popular media and news events, cross-curricular integration
- by posing and solving problems or asking questions about place, stories, and cultural practices

#### Communicating and representing

Explain and justify

- use mathematical arguments to convince
- includes anticipating consequences

- Have students explore which of two scenarios they would choose and then defend their choice.

- including oral, written, visual, use of technology
- communicating effectively according to what is being communicated and to whom

Represent

- using models, tables, graphs, words, numbers, symbols
- connecting meanings among various representations

Use mathematical vocabulary and language to contribute to discussions

- partner talks, small-group discussions, teacher-student conferences

Take risks when offering ideas in classroom discourse

- is valuable for deepening understanding of concepts
- can help clarify students’ thinking, even if they are not sure about an idea or have misconceptions

#### Connecting and reflecting

Reflect

- share the mathematical thinking of self and others, including evaluating strategies and solutions, extending, posing new problems and questions

Connect mathematical concepts

- to develop a sense of how mathematics helps us understand ourselves and the world around us (e.g., daily activities, local and traditional practices, popular media and news events, social justice, cross-curricular integration)

Use mistakes

- range from calculation errors to misconceptions

- by:
- analyzing errors to discover misunderstandings
- making adjustments in further attempts
- identifying not only mistakes but also parts of a solution that are correct

Incorporate

- by:
- collaborating with Elders and knowledge keepers among local First Peoples
- exploring the First Peoples Principles of Learning (http://www.fnesc.ca/wp/wp-content/uploads/2015/09/PUB-LFP-POSTER-Princi…; e.g., Learning is holistic, reflexive, reflective, experiential, and relational [focused on connectedness, on reciprocal relationships, and a sense of place]; Learning involves patience and time)
- making explicit connections with learning mathematics
- exploring cultural practices and knowledge of local First Peoples and identifying mathematical connections

- local knowledge and cultural practices that are appropriate to share and that are non-appropriated

- Bishop’s cultural practices: counting, measuring, locating, designing, playing, explaining (http://www.csus.edu/indiv/o/oreyd/ACP.htm_files/abishop.htm)
- Aboriginal Education Resources (www.aboriginaleducation.ca)
*Teaching Mathematics in a First Nations Context*, FNESC (http://www.fnesc.ca/resources/math-first-peoples/)